Vamp

What are the big differences?

I always thought that you hold a sword with your thumb and only top two fingers of your left hand right at the bottom. My friend who does Koga holds the sword with all his fingers with his left hand

Hikage

Ya know, it's hard to really pin-point something as precise as the holding of the sword. When I was first taught the sword, I learned to leave my bottom 2 fingers off the handle, and they rest on the pommel of the sword. Truth be told, that's not the way I do it now that I've had a chance to find what works for me. Ninjitsu is meant to be changed. If something does work for your particular personality or body type, change it in order that it does work. I don't know about you, but I'm not into shoving round pegs into square holes.
I'm not sure what you mean about the thumb, but if you mean only the index finger and thumb support the sword taht is right and wrong. You always want to have all your fingers on the sword as opposed to sticking out. This gives you more control and there are times when the back fingers come in handy. (get it, handy.. Anyway, it is also true that your index and thumb form a pivot point. When you are pivoting the sword in your hands, for example switching from a downward cut to an upward cut that pivot point is essential. Stephen Hayes has a great picture of how he holds the sword in one of his books. I wish I could remember what it was called (I had it, lent it out and never got it back). If it helps, it was black, rather tall, and has a picture on teh front of him with a sword of all things. he goes over some basic sword stances and cuts and that might better help answer your question.

-Hikage